#122589

order	512
nilpotency class	4
isoclinism	isoclinic class of #122544
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,3,8,6,11,5,10)(4,13,12,9,15,7,14,16)(17,18,21,26,22,25,34,41)(19,27,29,42,30,23,35,37)(20,24,32,40,33,39,45,48)(28,38,43,47,31,36,44,46);(2,7)(3,5)(4,14)(8,9)(10,16)(11,13)(12,15)(17,19,22,30)(18,23,25,27)(20,31,33,28)(21,29,34,35)(24,38,39,36)(26,42,41,37)(32,44,45,43)(40,46,48,47);(1,3,6,5)(2,8,11,10)(4,12,15,14)(7,16,13,9)(17,20,22,33)(18,24,25,39)(19,28,30,31)(21,32,34,45)(23,36,27,38)(26,40,41,48)(29,43,35,44)(37,46,42,47)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29785);